1. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH
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Derivative Formulas
General Rules
d d d
[ f (x) + g(x)] = f (x) + g (x) [ f (x) − g(x)] = f (x) − g (x) [c f (x)] = c f (x)
dx dx dx
d d d f (x) f (x)g(x) − f (x)g (x)
[ f (g(x))] = f (g(x))g (x) [ f (x)g(x)] = f (x)g(x) + f (x)g (x) =
dx dx dx g(x) [g(x)]2
Power Rules
d n d d d √ 1
(x ) = nx n−1 (c) = 0 (cx) = c ( x) = √
dx dx dx dx 2 x
Exponential
d x d x d d
[e ] = e x [a ] = a x ln a eu(x) = eu(x) u (x) er x = r er x
dx dx dx dx
Trigonometric
d d d
(sin x) = cos x (cos x) = −sin x (tan x) = sec2 x
dx dx dx
d d d
(cot x) = −csc2 x (sec x) = sec x tan x (csc x) = −csc x cot x
dx dx dx
Inverse Trigonometric
d 1 d 1 d 1
(sin−1 x) = √ (cos−1 x) = − √ (tan−1 x) =
dx 1 − x2 dx 1 − x2 dx 1 + x2
d 1 d 1 d 1
(cot−1 x) = − (sec−1 x) = √ (csc−1 x) = − √
dx 1 + x2 dx |x| x 2 − 1 dx |x| x 2 − 1
Hyperbolic
d d d
(sinh x) = cosh x (cosh x) = sinh x (tanh x) = sech2 x
dx dx dx
d d d
(coth x) = −csch2 x (sech x) = −sech x tanh x (csch x) = −csch x coth x
dx dx dx
Inverse Hyperbolic
d 1 d 1 d 1
(sinh−1 x) = √ (cosh−1 x) = √ (tanh−1 x) =
dx 1 + x2 dx x2 − 1 dx 1 − x2
d 1 d 1 d 1
(coth−1 x) = (sech−1 x) = − √ (csch−1 x) = − √
dx 1 − x2 dx x 1 − x2 dx |x| x 2 + 1
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2. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH
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Table of Integrals
Forms Involving a + bu
1 1
1. du = ln |a + bu| + c
a + bu b
u 1
2. du = 2 (a + bu − a ln |a + bu|) + c
a + bu b
u2 1
3. du = 3 [(a + bu)2 − 4a(a + bu) + 2a 2 ln |a + bu|] + c
a + bu 2b
1 1 u
4. du = ln +c
u(a + bu) a a + bu
1 b a + bu 1
5. du = 2 ln − +c
u 2 (a + bu) a u au
Forms Involving (a + bu)2
1 −1
6. du = +c
(a + bu)2 b(a + bu)
u 1 a
7. du = 2 + ln |a + bu| + c
(a + bu)2 b a + bu
u2 1 a2
8. du = 3 a + bu − − 2a ln |a + bu| + c
(a + bu)2 b a + bu
1 1 1 u
9. du = + 2 ln +c
u(a + bu)2 a(a + bu) a a + bu
1 2b a + bu a + 2bu
10. du = 3 ln − 2 +c
u 2 (a + bu)2 a u a u(a + bu)
Forms Involving a + bu
√ 2
11. u a + bu du = (3bu − 2a)(a + bu)3/2 + c
15b2
√ 2
12. u 2 a + bu du = (15b2 u 2 − 12abu + 8a 2 )(a + bu)3/2 + c
105b3
√ 2 2na √
13. u n a + bu du = u n (a + bu)3/2 − u n−1 a + bu du
b(2n + 3) b(2n + 3)
√
a + bu √ 1
14. du = 2 a + bu + a √ du
u u a + bu
√ √
a + bu −1 (a + bu)3/2 (2n − 5)b a + bu
15. du = − du, n = 1
u n a(n − 1) u n−1 2a(n − 1) u n−1
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3. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH
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√ √
1 1 a + bu − a
16a. √ du = √ ln √ √ + c, a > 0
u a + bu a a + bu + a
1 2 a + bu
16b. √ du = √ tan−1 + c, a < 0
u a + bu −a −a
√
1 −1 a + bu (2n − 3)b 1
17. √ du = − √ du, n = 1
u n a + bu a(n − 1) u n−1 2a(n − 1) u n−1 a + bu
u 2 √
18. √ du = 2 (bu − 2a) a + bu + c
a + bu 3b
u2 2 √
19. √ du = (3b2 u 2 − 4abu + 8a 2 ) a + bu + c
a + bu 15b3
un 2 √ 2na u n−1
20. √ du = u n a + bu − √ du
a + bu (2n + 1)b (2n + 1)b a + bu
Forms Involving a 2 + u 2, a > 0
√ √
21. a 2 + u 2 du = 1 u a 2 + u 2 + 1 a 2 ln u + a 2 + u 2 + c
2 2
√ √
22. u 2 a 2 + u 2 du = 1 u(a 2 + 2u 2 ) a 2 + u 2 − 1 a 4 ln u + a 2 + u 2 + c
8 8
√ √
a2 + u2 a+ a2 + u2
23. du = a 2 + u 2 − a ln +c
u u
√ √
a2 + u2 a2 + u2
24. du = ln u + a2 + u2 − +c
u2 u
1
25. √ du = ln u + a2 + u2 + c
a2 + u2
u2 1 1
26. √ du = u a 2 + u 2 − a 2 ln u + a2 + u2 + c
a2 + u2 2 2
1 1 u
27. √ du = ln √ +c
u a2 + u2 a a + a2 + u2
√
1 a2 + u2
28. √ du = − +c
u 2 a2 + u2 a2 u
Forms Involving a 2 ¯¯ u 2 , a > 0
√ u
29. a 2 − u 2 du = 1 u a 2 − u 2 + 1 a 2 sin−1 + c
2 2 a
√ u
30. u 2 a 2 − u 2 du = 1 u(2u 2 − a 2 ) a 2 − u 2 + 1 a 4 sin−1 + c
8 8 a
√ √
a2 − u2 a+ a2 − u2
31. du = a 2 − u 2 − a ln +c
u u
√ √
a2 − u2 a2 − u2 u
32. du = − − sin−1 + c
u2 u a
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4. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH
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1 u
33. √ du = sin−1 + c
a2 − u2 a
√
1 1 a+ a2 − u2
34. √ du = − ln +c
u a2 − u2 a u
u2 1 1 u
35. √ du = − u a 2 − u 2 + a 2 sin−1 + c
a2 − u2 2 2 a
√
1 a2 − u2
36. √ du = − +c
u2 a2 − u2 a2 u
Forms Involving u 2 ¯¯ a 2 , a > 0
√ √
37. u 2 − a 2 du = 1 u u 2 − a 2 − 1 a 2 ln u + u 2 − a 2 + c
2 2
√ √
38. u 2 u 2 − a 2 du = 1 u(2u 2 − a 2 ) u 2 − a 2 − 1 a 4 ln u + u 2 − a 2 + c
8 8
√
u2 − a2 |u|
39. du = u 2 − a 2 − a sec−1 +c
u a
√ √
u2 − a2 u2 − a2
40. du = ln u + u2 − a2 − +c
u2 u
1
41. √ du = ln u + u2 − a2 + c
u2 − a2
u2 1 1
42. √ du = u u 2 − a 2 + a 2 ln u + u2 − a2 + c
u2 − a2 2 2
1 1 |u|
43. √ du = sec−1 +c
u u2 − a2 a a
√
1 u2 − a2
44. √ du = +c
u2 u2 − a2 a2 u
Forms Involving 2au ¯¯ u 2
1 1 a−u
45. 2au − u 2 du = (u − a) 2au − u 2 + a 2 cos−1 +c
2 2 a
1 1 a−u
46. u 2au − u 2 du = (2u 2 − au − 3a 2 ) 2au − u 2 + a 3 cos−1 +c
6 2 a
√
2au − u 2 a−u
47. du = 2au − u 2 + a cos−1 +c
u a
√ √
2au − u 2 2 2au − u 2 a−u
48. du = − − cos−1 +c
u2 u a
1 a−u
49. √ du = cos−1 +c
2au − u 2 a
u a−u
50. √ du = − 2au − u 2 + a cos−1 +c
2au − u2 a
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5. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH
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u2 1 3 a−u
51. √ du = − (u + 3a) 2au − u 2 + a 2 cos−1 +c
2au − u 2 2 2 a
√
1 2au − u 2
52. √ du = − +c
u 2au − u2 au
Forms Involving sin u OR cos u
53. sin u du = −cos u + c
54. cos u du = sin u + c
55. sin2 u du = 1 u −
2
1
2 sin u cos u + c
56. cos2 u du = 1 u +
2
1
2 sin u cos u + c
57. sin3 u du = − 2 cos u −
3
1
3 sin2 u cos u + c
58. cos3 u du = 2
3 sin u + 1
3 sin u cos2 u + c
1 n−1
59. sinn u du = − sinn−1 u cos u + sinn−2 u du
n n
1 n−1
60. cosn u du = cosn−1 u sin u + cosn−2 u du
n n
61. u sin u du = sin u − u cos u + c
62. u cos u du = cos u + u sin u + c
63. u n sin u du = −u n cos u + n u n−1 cos u du + c
64. u n cos u du = u n sin u − n u n−1 sin u du + c
1
65. du = tan u − sec u + c
1 + sin u
1
66. du = tan u + sec u + c
1 − sin u
1
67. du = −cot u + csc u + c
1 + cos u
1
68. du = −cot u − csc u + c
1 − cos u
sin(m − n)u sin(m + n)u
69. sin(mu) sin(nu) du = − +c
2(m − n) 2(m + n)
sin(m − n)u sin(m + n)u
70. cos(mu) cos(nu) du = + +c
2(m − n) 2(m + n)
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6. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH
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cos(n − m)u cos(m + n)u
71. sin(mu) cos(nu) du = − +c
2(n − m) 2(m + n)
sinm−1 u cosn+1 u m−1
72. sinm u cosn u du = − + sinm−2 u cosn u du
m+n m+n
Forms Involving Other Trigonometric Functions
73. tan u du = −ln |cos u| + c = ln |sec u| + c
74. cot u du = ln |sin u| + c
75. sec u du = ln |sec u + tan u| + c
76. csc u du = ln |csc u − cot u| + c
77. tan2 u du = tan u − u + c
78. cot2 u du = −cot u − u + c
79. sec2 u du = tan u + c
80. csc2 u du = −cot u + c
81. tan3 u du = 1
2 tan2 u + ln |cos u| + c
82. cot3 u du = − 1 cot2 u − ln |sin u| + c
2
83. sec3 u du = 1
2 sec u tan u + 1
2 ln |sec u + tan u| + c
84. csc3 u du = − 1 csc u cot u +
2
1
2 ln |csc u − cot u| + c
1
85. tann u du = tann−1 u − tann−2 u du, n = 1
n−1
1
86. cotn u du = − cotn−1 u − cotn−2 u du, n = 1
n−1
1 n−2
87. secn u du = secn−2 u tan u + secn−2 u du, n = 1
n−1 n−1
1 n−2
88. cscn u du = − cscn−2 u cot u + cscn−2 u du, n = 1
n−1 n−1
1
89. du = 1 u ± ln |cos u ± sin u| + c
1 ± tan u 2
1
90. du = 1 u ∓ ln |sin u ± cos u| + c
1 ± cot u 2
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7. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH
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1
91. du = u + cot u ∓ csc u + c
1 ± sec u
1
92. du = u − tan u ± sec u + c
1 ± csc u
Forms Involving Inverse Trigonometric Functions
93. sin−1 u du = u sin−1 u + 1 − u2 + c
94. cos−1 u du = u cos−1 u − 1 − u2 + c
95. tan−1 u du = u tan−1 u − ln 1 + u 2 + c
96. cot−1 u du = u cot−1 u + ln 1 + u 2 + c
97. sec−1 u du = u sec−1 u − ln |u + u 2 − 1| + c
98. csc−1 u du = u csc−1 u + ln |u + u 2 − 1| + c
√
99. u sin−1 u du = 1 (2u 2 − 1) sin−1 u + 1 u 1 − u 2 + c
4 4
√
100. u cos−1 u du = 1 (2u 2 − 1) cos−1 u − 1 u 1 − u 2 + c
4 4
Forms Involving eu
1 au
101. eau du = e +c
a
1 1
102. ueau du = u− 2 eau + c
a a
1 2 2 2
103. u 2 eau du = u − 2u+ 3 eau + c
a a a
1 n au n
104. u n eau du = u e − u n−1 eau du
a a
1
105. eau sin bu du = (a sin bu − b cos bu)eau + c
a 2 + b2
1
106. eau cos bu du = (a cos bu + b sin bu)eau + c
a 2 + b2
Forms Involving ln u
107. ln u du = u ln u − u + c
108. u ln u du = 1 u 2 ln u − 1 u 2 + c
2 4
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8. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH
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1 1
109. u n ln u du = u n+1 ln u − u n+1 + c
n+1 (n + 1)2
1
110. du = ln |ln u| + c
u ln u
111. (ln u)2 du = u(ln u)2 − 2u ln u + 2u + c
112. (ln u)n du = u(ln u)n − n (ln u)n−1 du
Forms Involving Hyperbolic Functions
113. sinh u du = cosh u + c
114. cosh u du = sinh u + c
115. tanh u du = ln (cosh u) + c
116. coth u du = ln |sinh u| + c
117. sech u du = tan−1 |sinh u| + c
118. csch u du = ln |tanh 1 u| + c
2
119. sech2 u du = tanh u + c
120. csch2 u du = −coth u + c
121. sech u tanh u du = −sech u + c
122. csch u coth u du = −csch u + c
1
123. √ da = sinh−1 a + c
a2 + 1
1
124. √ da = cosh−1 a + c
a2 − 1
1
125. da = tanh−1 a + c
1 − a2
1
126. √ da = −csch−1 a + c
|a| a 2 + 1
1
127. √ da = −sech−1 a + c
a 1 − a2
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